Finite Group Databasebeta

← #30648401

Subgroups — #30648401

Order 16256.

484 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861232C211yesmaximal, characteristic, complemented
2C2×#16861215C211yesmaximal, characteristic, complemented
3#22550456C212yesmaximal, complemented
4#22550125C212yesmaximal, complemented
5#22550124C212yesmaximal, complemented
6#22550114C211yesmaximal, characteristic, complemented
7#22550113C211yesmaximal, characteristic, complemented
8#22550112C211yesmaximal, characteristic, complemented
9#107×D127C211yesmaximal, characteristic, complemented
10#22549876C212yesmaximal, complemented
11#467×C127C211yesmaximal, characteristic, complemented
12C2×#14523249C2212yes—
13C2×#14523248C2211yescharacteristic
14#16861232—24no—
15C2×#14523235C2212yes—
16C2×D1016C2211yescharacteristic
17#16861215—24no—
18C2×#14523232C2211yescharacteristic
19C2×#14523231C2211yescharacteristic
20C2×#14523231C2211yescharacteristic
21C2×C8×D127C2211yescharacteristic
22C2×C8×D127C2211yescharacteristic
23#16861187—22no—
24#16861185C2212yes—
25#16861184—22no—
26#16861184C2212yes—
27#16861183C2212yes—
28#16861182—22no—
29#16861182—22no—
30C4⋊C4×D127C2212yes—
31#16861143C2211yescharacteristic
32#16861142C2212yes—
33#16861133C2212yes—
34#16861131C2212yes—
35#16861129C2212yes—
36C2×Q16×C127C2211yescharacteristic
37C22×C8×C127C2211yescharacteristic
38C22⋊Q8×C127C2212yes—
39#107×C127C2211yescharacteristic
40#103×C127C2212yes—
41#14523249—24no—
42#14523249—24no—
43#14523248—22no—
44#14523248—22no—
45C2×Dic254D411yescharacteristic
46C2×D508—22no—
47C2×D508—22no—
48C2×D508D411yescharacteristic
49C2×D508C2312yes—
50C2×C22⋊D127—22no—
51C2×C22⋊D127D411yescharacteristic
52C2×C22⋊D127D411yescharacteristic
53C2×C4×D127—22no—
54C2×C4×D127—22no—
55C2×C4×D127C2311yescharacteristic
56C2×C4×D127C2311yescharacteristic
57#14523235—24no—
58D1016—22no—
59#14523232—22no—
60#14523231—22no—
61#14523231—22no—
62C2×C127⋊C8C2311yescharacteristic
63C2×C127⋊C8C2311yescharacteristic
64C8×D127—22no—
65C8×D127D414yescomplemented
66C4⋊Dic127C2312yes—
67Dic127⋊C4—22no—
68Dic127⋊C4—22no—
69(C2×D127)⋊C4—22no—
70(C2×D127)⋊C4—22no—
71(C2×D127)⋊C4—22no—
72C2×Q8×C127C2312yes—
73C22×C4×C127C2311yescharacteristic
74Q16×C127—22no—
75C2×C8×C127—22no—
76C2×C8×C127C2311yescharacteristic
77C2×C8×C127C2311yescharacteristic
78C4⋊C4×C127—22no—
79C4⋊C4×C127C2312yes—
80C22⋊C4×C127—22no—
81C22×D127—22no—
82C22×D127C2×D411yescharacteristic
83C22×D127C2×D411yescharacteristic
84Dic254—22no—
85D508—22no—
86D508—22no—
87D508—22no—
88D508—44no—
89D508—82no—
90C22⋊D127—22no—
91C22⋊D127—22no—
92C22⋊D127—82no—
93C2×Dic127—22no—
94C2×Dic127C2×D411yescharacteristic
95C2×Dic127C2×D411yescharacteristic
96C4×D127—21no—
97C4×D127—21no—
98C4×D127—44no—
99C4×D127—82no—
100C4×D127C2×D412yes—
101C4×D127#4112yes—
102C127⋊C8—21no—
103C127⋊C8C2×D412yes—
104C23×C127C2×D411yescharacteristic
105Q8×C127—22no—
106Q8×C127—22no—
107C2×C4×C127—21no—
108C2×C4×C127—21no—
109C2×C4×C127—22no—
110C2×C4×C127—22no—
111C2×C4×C127C2×D411yescharacteristic
112C2×C4×C127C2411yescharacteristic
113C8×C127—21no—
114C8×C127C2×D412yes—
115C22×C127—21no—
116C22×C127—21no—
117C22×C127—21no—
118C22×C127C22×D411yescharacteristic
119C2×D127—21no—
120C2×D127—21no—
121C2×D127—21no—
122C2×D127—21no—
123C2×D127—21no—
124C2×D127—42no—
125C2×D127—42no—
126C2×D127—42no—
127C2×D127#12112yes—
128C4×C127—21no—
129C4×C127—42no—
130C4×C127—42no—
131C4×C127C22×D411yescharacteristic
132C4×C127C2×#4111yescharacteristic
133Dic127—21no—
134Dic127—42no—
135Dic127#12112yes—
136C2×C127—41no—
137C2×C127C2×#12111yescharacteristic
138C2×C127#57611yescharacteristic
139C2×C127#57811yescharacteristic
140D127—22no—
141D127—41no—
142D127—82no—
143#3061—1271nomaximal, Sylow
144C127#306111yescharacteristic, complemented, Sylow
145C2×#135—1271no—
146C2×#131—1271no—
147#537—1272no—
148#468—1272no—
149#468—1272no—
150#467—1271no—
151#467D12711yescharacteristic, complemented
152#466—1271no—
153#466—1271no—
154C2×#107—1271no—
155#417—1272no—
156C2×#41—1271no—
157C2×#41—1272no—
158#135—2544no—
159C2×Q16C2×D12711yescharacteristic
160C2×SD16—1272no—
161C2×D8—1271no—
162#131—2544no—
163C2×M16—1271no—
164C2×M16—1271no—
165C2×M16—1271no—
166C22×C8—1271no—
167C22×C8—1271no—
168C22×C8C2×D12711yescharacteristic
169C22⋊Q8—1272no—
170C22⋊Q8—2542no—
171C22⋊Q8C2×D12712yes—
172#121—1272no—
173#121—1272no—
174#121—2542no—
175#121—2542no—
176#121—2542no—
177C2×C4⋊C4—1272no—
178#107—1271no—
179#107—1272no—
180#107C2×D12711yescharacteristic
181#103—1272no—
182#103C2×D12712yes—
183#102—1272no—
184#102—1272no—
185#41—2542no—
186#41—2542no—
187#41—2544no—
188#41—2544no—
189C2×Q8—1271no—
190C2×Q8C22×D12712yes—
191C2×D4—1271no—
192C2×D4—1271no—
193C2×D4—1271no—
194C2×D4—1272no—
195C2×D4—2542no—
196C2×D4—2542no—
197C2×D4—2542no—
198C22×C4—1271no—
199C22×C4—1271no—
200C22×C4—2542no—
201C22×C4—2542no—
202C22×C4C22×D12711yescharacteristic
203Q16—22no—
204SD16—2544no—
205D8—2542no—
206M16—2542no—
207M16—2542no—
208M16—2542no—
209C2×C8—22no—
210C2×C8—1271no—
211C2×C8—1274no—
212C2×C8—1271no—
213C2×C8—2542no—
214C2×C8C22×D12711yescharacteristic
215C2×C8C22×D12711yescharacteristic
216C4⋊C4—22no—
217C4⋊C4—1272no—
218C4⋊C4—2542no—
219C4⋊C4—2542no—
220C4⋊C4C22×D12712yes—
221C22⋊C4—22no—
222C22⋊C4—2542no—
223C22⋊C4—2542no—
224C22⋊C4—2542no—
225C23—1271no—
226C23—1271no—
227C23—2542no—
228C23D4×D12711yescharacteristic
229Q8—22no—
230Q8—22no—
231Q8—2542no—
232D4—10162no—
233D4—10162no—
234D4—5084no—
235D4—2542no—
236D4—2542no—
237D4—2542no—
238D4—2542no—
239D4—2542no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—1271no—
245C2×C4—1272no—
246C2×C4—1272no—
247C2×C4—1271no—
248C2×C4—10162no—
249C2×C4—5084no—
250C2×C4—2542no—
251C2×C4—2541no—
252C2×C4—2541no—
253C2×C4D4×D12711yescharacteristic
254C2×C4C23×D12711yescharacteristic
255C8—21no—
256C8—1272no—
257C8—2541no—
258C8D4×D12712yes—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—1272no—
263C22—5082no—
264C22—5082no—
265C22—5082no—
266C22—2541no—
267C22—2541no—
268C22—2541no—
269C22—2541no—
270C22—2541no—
271C22C2×D4×D12711yescharacteristic
272C4—21no—
273C4—42no—
274C4—42no—
275C4—1272no—
276C4—5082no—
277C4—2541no—
278C4C2×D4×D12711yescharacteristic
279C4#41×D12711yescharacteristic
280C2—41no—
281C2—10162no—
282C2—5081no—
283C2—2542no—
284C2#121×D12711yescharacteristic
285C2#2255067811yescharacteristic
286C2#2255068911yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#467×C127C2
socleC22×C127C22×D4
hypercenter#467D127
hypocenterC127#3061
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#16861232C2semidirect
2. C2×#16861215C2semidirect
3. #22550456C2semidirect
4. #22550125C2semidirect
5. #22550124C2semidirect
6. #22550114C2semidirect
7. #22550113C2semidirect
8. #22550112C2semidirect
9. #107×D127C2semidirect
10. #22549876C2semidirect
11. #467×C127C2semidirect
65. C8×D127D4semidirect
144. C127#3061semidirect
151. #467D127semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168612322
C2×#168612152
#225504562
#225501252
#225501242
#225501142
#225501132
#225501122
#107×D1272
#225498762
#467×C1272
#3061127